Suppose the interest rate is 8% APR with monthly compounding. What is the present value o an annuity that pays $90 every 6 months for 5 years?
This question is harder than it seems.
The problem is that the payment period does not coincide with the interest period.
So I will convert the 8% compounded monthly to a rate compounded semi-annually
let the semiannual rate be j
(1+j)^2 = (1.02)^4
1+j = (1.02)^2 = 1.0404
j = .0404
PV = 90(1 - 1.0404^-30)/.0404
= $ 1548.75
To calculate the present value of an annuity, we need to use the formula:
PV = C * (1 - (1 + r)^(-n)) / r
Where:
PV = Present Value
C = Cash Flow per period
r = interest rate per period
n = number of periods
In this case, the cash flow is $90, the interest rate is 8% APR (which needs to be converted to a monthly rate), and the number of periods is 5 years (10 six-month periods).
First, let's calculate the interest rate per period:
r = annual interest rate / number of compounding periods per year
r = 8% / 12 (since there are 12 months in a year)
r = 0.08 / 12
r = 0.00667 (rounded to 5 decimal places)
Now, plug in the values into the formula:
PV = $90 * (1 - (1 + 0.00667)^(-10)) / 0.00667
Calculating it step-by-step:
Step 1: Calculate the value inside the parentheses.
(1 + 0.00667)^(-10) ≈ 0.9389517
Step 2: Calculate (1 - (1 + 0.00667)^(-10))
1 - 0.9389517 ≈ 0.0610483
Step 3: Calculate the present value.
PV ≈ $90 * 0.0610483 / 0.00667
Step 4: Calculate the final result.
PV ≈ $824.44
Therefore, the present value of an annuity that pays $90 every 6 months for 5 years at an interest rate of 8% APR with monthly compounding, is approximately $824.44.
To calculate the present value of an annuity, we first need to understand the formula. The present value of an annuity is given by the formula:
PV = PMT x (1 - (1 + r)^(-n)) / r
Where:
PV = Present Value
PMT = Payment per period
r = Interest rate per period
n = Number of periods
In this case, the payment per period (PMT) is $90 every 6 months, the interest rate per period (r) is 8% APR (annual percentage rate) with monthly compounding, and the number of periods (n) is 5 years.
Let's break down the steps to calculate the present value:
Step 1: Calculate the interest rate per period (r).
Since the interest rate is given as an annual percentage rate, we need to convert it to a monthly rate. We can do this by dividing the annual rate by the number of compounding periods in a year. In this case, we have monthly compounding, so we divide the annual rate by 12.
Monthly interest rate = 8% / 12 = 0.08 / 12 = 0.0067 (rounded to 4 decimal places)
Step 2: Calculate the number of periods (n).
Since the annuity payments are made every 6 months and we're looking at a 5-year period, we have 5 years * 2 payments per year = 10 periods.
n = 10
Step 3: Calculate the present value (PV).
Using the formula:
PV = PMT x (1 - (1 + r)^(-n)) / r
Substituting the given values:
PV = $90 x (1 - (1 + 0.0067)^(-10)) / 0.0067
Now we can calculate this expression to find the present value of the annuity.